Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 6

Answer

\[ - \frac{{{{\cos }^3}x}}{3} + \frac{{{{\cos }^5}x}}{5} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\cos }^2}x{{\sin }^3}xdx} \hfill \\ \hfill \\ split\,\,{\sin ^3}x \hfill \\ \hfill \\ \int_{}^{} {{{\cos }^2}x{{\sin }^2}x\sin x\,dx} \hfill \\ \hfill \\ substitute\,\,{\sin ^2}x = 1 - {\cos ^2}x \hfill \\ \hfill \\ \int_{}^{} {{{\cos }^2}x\,\left( {1 - {{\cos }^2}x} \right)\sin xdx} \hfill \\ \hfill \\ Distribute \hfill \\ \hfill \\ \int_{}^{} {\,\left( {{{\cos }^2}x\sin x - {{\cos }^4}x\sin x} \right)dx} \hfill \\ \hfill \\ rewrite \hfill \\ \hfill \\ - \int_{}^{} {{{\cos }^2}x\,\left( { - \sin x} \right)dx + \int_{}^{} {{{\cos }^4}x\,\left( { - \sin x} \right)dx} } \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ - \frac{{{{\cos }^3}x}}{3} + \frac{{{{\cos }^5}x}}{5} + C \hfill \\ \hfill \\ \end{gathered} \]
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